Grade 1: Unit 1.MD.C.4 Represent and interpret data

Overview: The overview statement is intended to provide a summary of major themes in this unit.

In this unit, students begin to organize and represent categorical data. For example, if a collection of specimens is sorted into two piles based on which specimens have wings and which do not, students might present the two piles of specimens on a piece of paper, by making a group of marks for each pile, as shown below (the marks could also be circles, for example). The groups of marks should be clearly labeled to reflect the attribute in question.

The work shown in the figure is the result of an intricate process. At first, we have before us a jumble of specimens with many attributes. Then there is a narrowing of attention to a single attribute (wings or not). Then the objects might be arranged into piles. The arranging of objects into piles is then mirrored in the arranging of marks into groups. In the end, each mark represents an object; its position in one column or the other indicates whether or not that object has a given attribute.

There is no single correct way to represent categorical data – and the Standards do not require Grade 1 students to use any specific format. However, students should be familiar with mark schemes like the one show in the figure. Another format that might be useful in Grade 1 is a picture graph in which one picture represents one object. (Note that picture graphs are not an expectation in the Standards until Grade 2) If different students devise different ways to represent the same data set, then the class might discuss relative strengths and weaknesses of each scheme (SMP5)

Students’ data work in Grade 1 has important connections to addition and subtraction, as noted in Table 1 on page 4 of the K-3, Categorical Data Progression. Students in Grade 1 can ask and answer questions about categorical data based on a representation of the data. For example, with reference to the figure above, a student might ask how many specimens there were altogether, representing this problem by writing an equation such as 7 + 8 = ?. Students can also ask and answer questions leading to other kinds of addition and subtraction problems (1.OA), such as compare problems or problems involving the addition of three numbers for situations with three categories.

*Overview taken from the “Progressions for the Common Core State Standards in Mathematics – K-3 Categorical Data.Teacher Notes: The information in this component provides additional insights which will help the educator in the planning process for the unit. July 27, 2013 Page 1 of 19

Grade 1: Unit 1.MD.C.4 Represent and interpret data

Review the Progressions for K-3, Categorical Data; Grades 2-5 Measurement Data at: http://commoncoretools.me/wp-content/uploads/2011/06/ccss_progression_md_k5_2011_06_20.pdf to see the development of the understanding of the interpretation and display of both categorical and measurement data as stated by the Common Core Standards Writing Team, which is also the guiding information for the PARCC Assessment development.

When implementing this unit, be sure to incorporate the Enduring Understandings and Essential Questions as the foundation for your instruction, as appropriate.

Students should engage in well-chosen, purposeful, problem-based tasks. A good mathematics problem can be defined as any task or activity for which the students have no prescribed or memorized rules or methods, nor is there a perception by students that there is a specific correct solution method (Hiebert et al., 1997). A good mathematics problem will have multiple entry points and require students to make sense of the mathematics. It should also foster the development of efficient computations strategies as well as require justifications or explanations for answers and methods.

It is important to incorporate addition and subtraction questions and problems related to the data being discussed in order to make the connection between the two domains of mathematics. Students need to see the need to apply one set of mathematical knowledge when learning about another.

Enduring Understandings: Enduring understandings go beyond discrete facts or skills. They focus on larger concepts, principles, or processes. They are transferable and apply to new situations within or beyond the subject.

Data can be sorted and organized in order to answer questions and solve problems. The quality of the question used impacts the data collected and the validity of the results. The way data is displayed or organized influences interpretation.

Essential Questions: A question is essential when it stimulates multi-layered inquiry, provokes deep thought and lively discussion, requires students to consider alternatives and justify their reasoning, encourages re-thinking of big ideas, makes meaningful connections with prior learning, and provides students with opportunities to apply problem-solving skills to authentic situations.

Why is data collected and analyzed? How can information be gathered, recorded, and organized? How do people use data to influence others? How can predictions be made based on data? How does the type of data influence the choice of display? How does the way we display data influence our interpretation of it? How does collecting data help us solve problems or make decisions in our world? Can data be sorted or organized in different ways? Is one way better than another?

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Grade 1: Unit 1.MD.C.4 Represent and interpret data

Content Emphasis by Cluster in Grade 1: According to the Partnership for the Assessment of Readiness for College and Careers (PARCC), some clusters require greater emphasis than others. The table below shows PARCC’s relative emphasis for each cluster. Prioritization does not imply neglect or exclusion of material. Clear priorities are intended to ensure that the relative importance of content is properly attended to. Note that the prioritization is in terms of cluster headings.

Key: Major Clusters Supporting Clusters○ Additional Clusters

Operations and Algebraic Thinking

Represent and solve problems involving addition and subtraction. Understand and apply properties of operations and the relationship between addition and subtraction. Add and subtract within 20. Work with addition and subtraction equations.

Number and Operations in Base Ten

Extend the counting sequence. Understand place value. Use place value understanding and properties of operations to add and subtract.

Measurement and Data

Measure lengths indirectly and by iterating length units.○ Tell time and write time Represent and interpret data.

Geometry

○ Reason with shapes and their attributes.

Focus Standards: (Listed as Examples of Opportunities for In-Depth Focus in the PARCC Content Framework documents for Grades 3-8)

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Grade 1: Unit 1.MD.C.4 Represent and interpret data

According to the Partnership for the Assessment of Readiness for College and Careers (PARCC), this component highlights some individual standards that play an important role in the content of this unit. Educators from the State of Maryland have identified the following Standards as Focus Standards. Should PARCC release this information for Prekindergarten through Grade 2, this section would be updated to align with their list. Educators may choose to give the indicated mathematics an especially in-depth treatment, as measured for example by the number of days; the quality of classroom activities for exploration and reasoning, the amount of student practice, and the rigor of expectations for depth of understanding or mastery of skills.

1.OA.A.1 Use addition and subtraction within 20 to solve word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem.

1.OA.A.2 Solve word problems that call for addition of three whole numbers whose sum is less than or equal to 20, e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem.

Possible Student Outcomes: The following list provides outcomes that describe the knowledge and skills that students should understand and be able to do when the unit is completed. The outcomes are often components of more broadly-worded standards and sometimes address knowledge and skills necessarily related to the standards. The lists of outcomes are not exhaustive, and the outcomes should not supplant the standards themselves. Rather, they are designed to help teachers delve deeply into the standards and augment as necessary, providing added focus and clarity for lesson planning purposes. This list is not intended to imply any particular scope or sequence.

The student will: Sort objects or pictures into two or three categories based on one common attribute. Record sorted categories using marks, stamps, picture graphs, etc. with each symbol used representing one data object each. Answer questions about the sorted sets such as: Which has more? Which has less? How many are their all together? Solve simple put-together, take-apart, and compare problems using the information represented in the sorted sets.

Progressions from Common Core State Standards in Mathematics: For an in-depth discussion of the overarching, “big picture” perspective on student learning of content related to this unit, see:

The Progressions for K-3, Categorical Data; Grades 2-5 Measurement Data at: http://commoncoretools.me/wp-content/uploads/2011/06/ccss_progression_md_k5_2011_06_20.pdf to see the development of the understanding of the interpretation and display of both categorical and measurement data as stated by the Common Core Standards Writing Team, which is also the guiding information for the PARCC Assessment development.

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Grade 1: Unit 1.MD.C.4 Represent and interpret data

Vertical Alignment: Vertical curriculum alignment provides two pieces of information: (1) a description of prior learning that should support the learning of the concepts in this unit, and (2) a description of how the concepts studied in this unit will support the learning of additional mathematics.

Key Advances from Previous Grades:

○ Students in Prekindergarten:- Directly compare two objects with a measurable attribute in common, using words such as longer/shorter, heavier/lighter, or

taller/shorter.- Sort objects into self-selected and given categories.- Compare categories using words such as more or same.

o Students in Kindergarten:- Directly compare two objects with a measurable attribute in common, to see which object has “more of”/”less of” the

attribute, and describe the difference. For example, directly compare the heights of two children and describe one as taller/shorter.

- Classify objects into given categories; count the numbers of objects in each category and sort the categories by count.

Additional Mathematics:

○ Students in Grade 2:- Measure related objects to collect and graph measurement data.- Answer questions about the data displayed in a graph.- Create a picture graph with a single-unit scale to represent a data set with up to four categories.- Create a bar graph with a single-unit scale to represent a data set with up to four categories.- Solve simple put-together, take-apart, and compare problems using information presented in a bar graph.

o Students in Grades 3:- Draw a scaled picture graph and a scaled bar graph to represent a data set with several categories. Solve-one- and two-

step “how many more” and “how many less” problems using information presented in scaled bar graphs.- Generate measurement data by measuring lengths using rulers marked with halves and fourths of an inch. Show the data

by making a line plot, where the horizontal scale is marked off in appropriate units – whole numbers, halves, or quarters.

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Grade 1: Unit 1.MD.C.4 Represent and interpret data

o Students in Grades 4:- Make a line plot to display a data set of measurements in fractions of a unit (1/2, ¼, 1/8). Solve problems involving

addition and subtraction of fractions by using information presented in line plots.

o Students in Grades 5:- Make a line plot to display a data set of measurements in fractions of a unit (1/2, ¼, 1/8). Use operations on fractions for

this grade to solve problems involving information presented in line plots.

o Students in Grades 6:- Develop understanding of statistical variability.- Summarize and describe distributions.- Use ran

o Students in Grades 7:- Use random sampling to draw inferences about a population.- Draw informal comparative inferences about two populations.- Investigate chance processes and develop, use, and evaluate probability models.

o Students in Grades 8:- Investigate patterns of association in bivariate data.

Possible Organization of Unit Standards: This table identifies additional grade-level standards within a given cluster that support the over-arching unit standards from within the same cluster. The table also provides instructional connections to grade-level standards from outside the cluster.

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Grade 1: Unit 1.MD.C.4 Represent and interpret data

Connections to the Standards for Mathematical Practice: This section provides examples of learning experiences for this unit that support the development of the proficiencies described in the Standards for Mathematical Practice. These proficiencies correspond to those developed through the Literacy Standards. The statements provided offer a few examples of connections between the Standards for Mathematical Practice and the Content Standards of this unit. The list is not exhaustive and will hopefully prompt further reflection and discussion.

In this unit, educators should consider implementing learning experiences which provide opportunities for students to:

1. Make sense of problems and persevere in solving them.

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Over-Arching Standards

Supporting Standards within the Cluster

Instructional Connections outside the Cluster

1.MD.C.4 Organize, represent, and interpret data with up to three categories; ask and answer questions about the total number of data points, how many in each category, and how many more or less are in one category than in another.

1.OA.A.1 Use addition and subtraction within 20 to solve word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem.

1.OA.A.2 Solve word problems that call for addition of three whole numbers whose sum is less than or equal to 20, e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem.

1.OA.D.7 Understand the meaning of the equal sign, and determine if equations involving addition and subtraction are true or false. For example, which of the following equations are true and which are false? 6 = 6, 7 = 8 – 1, 5 + 2 = 2 + 5, 4 + 1 = 5 + 2.

Grade 1: Unit 1.MD.C.4 Represent and interpret data

a. Determine what the problem is asking for: sorting of data into categories, comparison of categories, total number of data items, etc.b. Determine whether concrete or virtual models, pictures, mental mathematics, or equations are the best tools for solving the problem.c. Check the solution with the problem to verify that it does answer the question asked.

2. Reason abstractly and quantitativelya. Compare the different categories to determine which has more or less than the other.b. Justify why their interpretation of the data is correct using specific examples from the categories.

3. Construct Viable Arguments and critique the reasoning of others.a. Compare the different categories to determine which has more or less than the other. b. Examine the steps taken that produce an incorrect response and provide a viable argument as to why the process produced an

incorrect response.c. Use the calculator to verify the correct solution, when appropriate.

4. Model with Mathematicsa. Construct visual models using concrete or virtual manipulatives, pictures, or equations to justify thinking and display the solution.b. Record with strike marks, stickers, or symbols the number of objects in each category.c. Display the data in a picture graph in which each symbol represents only one piece of data.

5. Use appropriate tools strategicallya. Use concrete manipulatives, tally charts, or other models, as appropriate.b. Use the calculator to verify computation.

6. Attend to precisiona. Use mathematics vocabulary such as data, picture graph, category, etc. properly when discussing problems.b. Demonstrate understanding of the mathematical processes required to solve a problem by carefully showing all of the steps in the

solving process.c. Correctly write and read equations.d. Use <, =, and > appropriately to compare expressions.

7. Look for and make use of structure.a. Use the patterns in your displays to make sense of the comparisons.b. Use the relationships demonstrated in the organization of the data within the categories to compare their values.

8. Look for and express regularity in reasoninga. Use the patterns illustrated in the grouping of data items to compare the different categories.b. Use the relationships demonstrated in the between the categories to state generalizations about the data.

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Grade 1: Unit 1.MD.C.4 Represent and interpret data

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Grade 1: Unit 1.MD.C.4 Represent and interpret data

Content Standards with Essential Skills and Knowledge Statements and Clarifications: The Content Standards and Essential Skills and Knowledge statements shown in this section come directly from the Maryland State Common Core Curriculum Frameworks. Clarifications were added as needed. Educators should be cautioned against perceiving this as a checklist. All information added is intended to help the reader gain a better understanding of the standards.

Standard Essential Skills and Knowledge ClarificationStandard: 1.MD.4Organize, represent, and interpret data with up to three categories; ask and answer questions about the total number of data points, how many in each category, and how many more or less are in one category than in another.

Essential Skills and Knowledge Ability to sort data into

separate categories Ability to display data in

appropriate graph, such as a picture graph

Ability to answer questions about the data such as ‘Which category has more?’ ‘Which category has less?’ ‘What is the favorite snack of our class?’ ‘How many more stickers does Sam have than John?’

Students create object graphs and tally charts using data relevant to their lives (e.g., favorite ice cream, eye color, pets, etc.). Graphs may be constructed by groups of students as well as by individual students.

Counting objects should be reinforced when collecting, representing, and interpreting data. Students describe the object graphs and tally charts they create. They should also ask and answer questions based on these charts or graphs that reinforce other mathematics concepts such as sorting and comparing. The data chosen or questions asked give students opportunities to reinforce their understanding of place value, identifying ten more and ten less, relating counting to addition and subtraction and using comparative language and symbols.

Students may use an interactive whiteboard to place objects onto a graph. This gives them the opportunity to communicate and justify their thinking.

Example: Students pose a question and the 3 possible responses.Which is your favorite flavor of ice cream? Chocolate, vanilla or strawberry? Students collect their data by using tallies or another way of keeping track.Students organize their data by totaling each category in a chart or table:

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Grade 1: Unit 1.MD.C.4 Represent and interpret data

Standard Essential Skills and Knowledge ClarificationWhat is your favorite flavor of ice cream?Chocolate 12Vanilla 5Strawberry 6

Students interpret the data by comparing categories: What does the data tell us? Does it answer our question?- More people like chocolate than the other two

flavors.- Only 5 people like vanilla.- Six people liked strawberry.- 7 more people liked chocolate than vanilla.- The number of people that like vanilla was 1 less

than the number of people who liked strawberry.- The number of people who liked either vanilla or

strawberry was 1 less than the number of people who liked chocolate.

- 23 people answered this question.-

(Taken from the Arizona Mathematics Standards and the Kansas Mathematics Standards.)

Evidence of Student Learning: The Partnership for the Assessment of Readiness for College and Careers (PARCC) has awarded the Dana Center a grant to develop the information for this component. This information will be provided at a later date. The Dana Center, located at the University of Texas in Austin, encourages high academic standards in mathematics by working in partnership with local, state, and national education entities.  Educators at the Center collaborate with their partners to help school systems nurture students' intellectual passions.  The Center advocates for every student leaving school prepared for success in postsecondary education and in the contemporary workplace.

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Grade 1: Unit 1.MD.C.4 Represent and interpret data

Fluency Expectations and Examples of Culminating Standards: This section highlights individual standards that set expectations for fluency, or that otherwise represent culminating masteries. These standards highlight the need to provide sufficient supports and opportunities for practice to help students meet these expectations. Fluency is not meant to come at the expense of understanding, but is an outcome of a progression of learning and sufficient thoughtful practice. It is important to provide the conceptual building blocks that develop understanding in tandem with skill along the way to fluency; the roots of this conceptual understanding often extend one or more grades earlier in the standards than the grade when fluency is finally expected.

Fluently add and subtraction within 10.

Common Misconceptions: This list includes general misunderstandings and issues that frequently hinder student mastery of concepts regarding the content of this unit.

Incorrectly recording the data from the table to the picture graph or sorting categories. Forgetting to apply one-to-one correspondence when comparing the data from different categories. Incorrectly sorting the data into categories.

Interdisciplinary Connections: Interdisciplinary connections fall into a number of related categories:

Literacy standards within the Maryland Common Core State Curriculum Science, Technology, Engineering, and Mathematics standards Instructional connections to mathematics that will be established by local school systems, and will reflect their specific grade-level

coursework in other content areas, such as English language arts, reading, science, social studies, world languages, physical education, and fine arts, among others.

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Grade 1: Unit 1.MD.C.4 Represent and interpret data

Available Model Lesson Plan(s)

The lesson plan(s) have been written with specific standards in mind.  Each model lesson plan is only a MODEL – one way the lesson could be developed.  We have NOT included any references to the timing associated with delivering this model.  Each teacher will need to make decisions related to the timing of the lesson plan based on the learning needs of students in the class. The model lesson plans are designed to generate evidence of student understanding.

This chart indicates one or more lesson plans which have been developed for this unit. Lesson plans are being written and posted on the Curriculum Management System as they are completed. Please check back periodically for additional postings.

Standards Addressed Title Description/Suggested Use

1.MD.C.4Data Helps Us Answer Everyday

Questions

Students will sort data into three categories and record information using a symbol to represent each student choice. They will represent data in an appropriate graph such as a picture graph and answer questions about the data such as, “Which category has more?” “Which category has less?” “How do cafeteria workers use the data we collect?” Students will solve simple put-together, take-apart, and compare problems using the information represented in the sorted sets.

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Grade 1: Unit 1.MD.C.4 Represent and interpret data

Available Lesson SeedsThe lesson seed(s) have been written with specific standards in mind.  These suggested activity/activities are not intended to be prescriptive, exhaustive, or sequential; they simply demonstrate how specific content can be used to help students learn the skills described in the standards. Seeds are designed to give teachers ideas for developing their own activities in order to generate evidence of student understanding.

This chart indicates one or more lesson seeds which have been developed for this unit. Lesson seeds are being written and posted on the Curriculum Management System as they are completed. Please check back periodically for additional postings.

Standards Addressed Title Description/Suggested Use

1.MD.C.4 First Bite

After listening to The Gingerbread Man, students either take a first bite of a Gingerbread Man cookie or tear off one piece of a paper Gingerbread Man. They record the data based on their ‘first bites’ and determine which part was the most popular piece bitten off.

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Grade 1: Unit 1.MD.C.4 Represent and interpret data

Sample Assessment Items: The items included in this component will be aligned to the standards in the unit and will include: Items purchased from vendors PARCC prototype items PARCC public released items Maryland Public release items Formative Assessment

Topic Standards Addressed

Link Notes

Favorite Ice Cream Flavor 1.MD.C.4

http://www.illustrativemathematics.org/illustrations/506

In this task entitled ‘Favorite Ice Cream Flavor’, students collect data about which type of ice cream is the most popular and which is the least popular.

Weather Graph Data 1.MD.C.4 http://www.illustrativemathematics.org/

illustrations/1233

The purpose of this task is for students to represent and interpret weather data, as well as answer questions about the data they have recorded. Teachers could modify this task to include the most common weather in their local area.

Interventions/Enrichments: (Standard-specific modules that focus on student interventions/enrichments and on professional development for teachers will be included later, as available from the vendor(s) producing the modules.)

Vocabulary/Terminology/Concepts: This section of the Unit Plan is divided into two parts. Part I contains vocabulary and terminology from standards that comprise the cluster which is the focus of this unit plan. Part II contains vocabulary and terminology from standards outside of the focus cluster. These “outside standards” provide important instructional connections to the focus cluster.

Part I – Focus Cluster:visual representations of numerals: concrete materials or pictures that represent specific numerals, showing the quantity represented by those

numerals. Examples:

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1 2 3 4 5

2 3 4 5

Grade 1: Unit 1.MD.C.4 Represent and interpret data

Part II – Instructional Connections outside the Focus Cluster

Counting All: the very first addition counting strategy in which a student counts all of the objects, pictures, or items in a problem to determine the total and solve the problem. This is the least efficient counting strategy to use and should lead to the more efficient Counting On strategies. Example: Bobby has two counters and Susie has three. How many do they have all together?

Counting On: an addition counting strategy in which a student starts with one set of objects and counts up to solve the problem. Example: Bobby has two counters and Susie has three. How many do they have all together?

Counting On from the Larger Number: an addition counting strategy in which a student starts with the largest set of objects and counts up to solve the problem. Example: Bobby has two counters and Susie has three. How many do they have all together?

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1 2 3 4 5 6 7 8 9 10

Grade 1: Unit 1.MD.C.4 Represent and interpret data

Counting Up: a subtraction counting strategy in which a student counts up from one part to the whole in order to find the missing part. Example: 9 – 6 = ? The student would count starting at 6, saying “7, 8, 9” determining that, by counting up three numbers, the missing part of the number sentence is ‘3’.

Resources: Free Resources:

http://wps.ablongman.com/ab_vandewalle_math_6/0,12312,3547876-,00.html Reproducible blackline masters http://lrt.ednet.ns.ca/PD/BLM_Ess11/table_of_contents.htm mathematics blackline masters http://yourtherapysource.com/freestuff.html Simple activities to encourage physical activity in the classroom http://www.mathsolutions.com/index.cfm?page=wp9&crid=56 Free lesson plan ideas for different grade levels http://sci.tamucc.edu/~eyoung/literature.html links to mathematics-related children’s literature http://www.nctm.org/ National Council of Teachers of Mathematics www.k-5mathteachingresources.com Extensive collection of free resources, math games, and hands-on math activities aligned with the

Common Core State Standards for Mathematics http://elementarymath.cmswiki.wikispaces.net/Standards+for+Mathematical+Practice Common Core Mathematical Practices in Spanish http://mathwire.com/ Mathematics games, activities, and resources for different grade levels http://www.pbs.org/teachers/math/ interactive online and offline lesson plans to engage students. Database is searchable by grade level and

content http://www.k8accesscenter.org/training_resources/MathWebResources.asp valuable resource including a large annotated list of free web-

based math tools and activities. http://www.cast.org/udl/index.html Universal Design for Learning

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Grade 1: Unit 1.MD.C.4 Represent and interpret data

http://engageny.org/wp-content/uploads/2012/05/Shifts-for-Students-and-Parents.pdf Information for parents and students about the Shifts associated with the CCSS.

http://havefunteaching.com/ Various resources, including tools such as sets of Common Core Standards posters. http://michellef.essdack.org/links Numerous mathematics links. http://illustrativemathematics.org/ Tasks that align with the MD CCSS.\ http://www.aimsedu.org/Puzzle/categories/topological.html Puzzles to challenge students of various ages. http://www.insidemathematics.org/index.php/home Mathematics resources for teachers. http://www.illustrativemathematics.org/illustrations/506 sample task for collecting and comparing data. http://www.azed.gov/standards-practices/mathematics-standards/ Arizona Mathematics Standards http://www.ksde.org/Default.aspx?tabid=5646 Kansas Mathematics Flipbooks

Math Related Literature:

St. George, Judith So You Want to Be President?

References: ------. 2000. Principles and Standards for School Mathematics. Reston, VA: National Council of Teachers of Mathematics.

Arizona Department of Education. “Arizona Academic content Standards.” Web. 28 June 2010 http://www.azed.gov/standards-practices/common-standards/

Bamberger, H.J., Oberdorf, C., Schultz-Ferrell, K. (2010). Math Misconceptions: From Misunderstanding to Deep Understanding.

Burns, M. (2007 ) About Teaching Mathematics: A K-8 Resource. Sausalito, CA: Math Solutions Publications.

Christinson, J., Wiggs, M.D., Lassiter, C.J., Cook, L. (2012). Navigating the Mathematics Common Core State Standards: Getting Ready for the Common Core Handbook Series, Book 3. Englewood, CO: Lead+, Learn, Press, a Division of Houghton Mifflin Harcourt.

Copley, J. (2010). The Young Child and Mathematics. Reston, VA: National Council of Teachers of Mathematics.

The Common Core Standards Writing Team (2011). Progressions for the Common Core State Standards in Mathematics (draft), accessed at:

Cross, C.T.,Woods, T. A., & Schweingruber, H. (2009). Mathematics Learning in Early Childhood: Paths Toward Excellence and Equity. Washington, D.C.: National Research Council of the National Academies

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North Carolina Department of Public Instruction. Web. February 2012. North Carolina Department of Public Instruction. Web. February 2012 http://www.ncpublicschools.org/acre/standards/common-core-tools/#unmath

Van de Walle, J. A., Lovin, J. H. (2006). Teaching Student-Centered mathematics, Grades K-3. Boston, MASS: Pearson Education, Inc.

Arizona Mathematics Standards http://www.azed.gov/standards-practices/mathematics-standards/

Kansas Mathematics Standards Flipbooks http://www.ksde.org/Default.aspx?tabid=5646

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